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Interpretation of Resonant and Non-Resonant
Beat-Wave Excitation: Experiments and
Simulations
C. V. Filip, S. Ya. Tochitsky, R. Narang, C. E. Clayton, K. A. Marsh,
and C. Joshi
Electrical Engineering Department, University of California at Los Angeles, 405 Hilgard Avenue,
Los Angeles, California 90095
Abstract. Beat-wave excited relativistic plasma waves (RPWs) are investigated using collinear
Thomson scattering. The RPWs are driven by a two-wavelength CO2 laser pulse with a peak power
of ~1TW which also produces plasmas through tunneling ionization of He, Ar and H2. The beat
frequency of the CO2 pulse is kept constant while the plasma density is changed from 1015cm~3 to
1017cm~3. RPWs are thus excited both at resonance, around 1016cm~3, and off-resonance. The plasma
waves are sampled with a 532-nm probe pulse. The scattered light, comprised of red and blue
shifted sidebands, is frequency and time resolved with a spectrometer and a streak camera. The
amplitude of the RPWs is found from the amount of light scattered into the sidebands. We compare
the experimental results with 2-D PIC simulations. Furthermore, simulations are used to study and
compare the RPWs excited in resonant and non-resonant conditions.
INTRODUCTION
Acceleration of electrons in plasmas with gradients in excess of 1 GeV/m is
possible through the excitation of RPWs using lasers, a method that was suggested by
Tajima and Dawson1 in 1979. In this paper it has been shown that a relatively long,
two-frequency laser pulse can efficiently excite a relativistic plasma wave. The
ponderomotive force associated with the beat pattern of the laser pulse bunches the free
electrons in plasma in such a way that a relativistic, forward-moving longitudinal wave
will develop. Externally injected electrons or trapped plasma electrons that are placed
at the proper phase will "ride" this wave and get accelerated with gradients 10 to 100
times larger than those provided by the conventional accelerators. This method
designed to accelerate electrons is called the plasma beat-wave acceleration (PBWA)
technique.
A condition for efficient excitation of the plasma wave is that the frequency
difference Aco=co1-C02, where cOj and co2 are the laser frequencies, equals the plasma
frequency, cop. Excitation of RPW in this regime is called resonant excitation and
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
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Aco=cop is the resonant condition. When co1~co2=co»cop, the phase velocity of the
plasma wave, vp=c[l-(cOp/co)2]1/2, is close to the speed of light, which makes
acceleration of relativistic electrons possible.
Previous experiments have studied the beat-wave excitation of RPWs when the
resonant condition is satisfied.2"4 In our experiment Aco is maintained constant while cop
is varied over a wide range, from approximately 0.5 Aco up to 4 Aco. RPWs are thus
excited at resonance and off-resonance. In all cases Thomson scattering (TS) was the
main diagnostic for the plasma wave.
The main result from the TS measurements is the amplitude of the RPW, 8, for
each particular plasma density. This information is also extracted from 2-D PIC
simulations that are performed to model the beat-wave excitation process. The output
of the code is compared to the experimental results. Furthermore, simulations are used
to study the similarities and differences between excitation of RPWs at resonance and
off-resonance.
COLLINEAR THOMSON SCATTERING
Thomson scattering is a well-known technique used to characterize laboratory
plasmas. The scattering process represents the interaction between the incident probe
laser pulse and the plasma electrons. Each electron scatters the light according to its
individual motion. The light waves scattered from each electron add up to construct the
total scattered beam that holds valuable information about the plasma.
The TS process is governed by two conservation laws. The first is the conservation
of momentum, which leads to:
Ks=Kpr—*^epw>
\ I/
where ks, kpr and kepw are the wave vectors of the scattered pulse, incident probe beam
and electron plasma wave, respectively. The conservation of energy or the frequency
matching condition, is related to the scattered light that has a frequency shifted from
the probe frequency by an amount equal to the frequency of the plasma fluctuation
(<oepw):
cos=copr±coepw,
(2)
where cos and copr are the frequencies of the scattered and probe electromagnetic waves,
respectively. Both of these equalities are exactly satisfied for RPWs only for the
collinear scattering geometry where the probe pulse propagates on the same direction
with the plasma wave.
Historically, for the CO2 driven PBWA case, the longitudinal component of the
plasma wave was first detected through self-scattering of the laser pulse that is driving
the plasma wave.2'5 In addition, the transverse component of the RPW was recorded
using an independent probe pulse produced by a ruby laser.2 In these experiments, the
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plasma waves were excited at the resonance condition when the plasma density
n~1016 cm"3. Another set of PBWA experiments at resonance were done using 1-um
pulses produced by solid-state lasers at n=1017 cm"3. In these measurements, performed
at higher plasma densities, the collinear TS technique was successfully used to record
the RPWs with an independent probe pulse.3'4
The novelty in our experiment is the probing of RPWs driven at around 1016 cm"3 in
a collinear geometry. An independent probe pulse, weak enough not to affect the
plasma or the plasma wave, is used to sample the RPW. In the same time, due to the
extremely low efficiency of the TS process, the probe pulse is sufficiently intense to
bring the scattered light at detectable levels.
The ratio of the scattered power Ps, over the incident power of the probe beam, P0,
is given by:6
PyP 0 =0.25(5nr 0 ^ p L z ) 2 ,
(3)
3
where Sn (cm" ) represents the density of the fluctuation within the plasma, r0 (cm) is
the classical electron radius, ?lp (cm) is the wavelength of the probe laser, and Lz (cm) is
the interaction length between the electron wave and the probing light. For example, in
a typical Thomson scattering experiment on a 10% plasma wave at n=1016 cm"3 having
Sn=1015 cm"3, 2tp=5 10"5 cm, and Lz=0.1 cm the ratio PJ P0 becomes -5 10"8. In addition,
the complicated nature of the experimental setup induces approximately 10 times
losses at all wavelengths. The power of the scattered light, Ps, thus becomes
approximately 10~8-10~10 of the incident power of the probe pulse.
The very small Thomson scattering efficiency is not the only difficult part of this
experiment. If we consider T| the index of refraction of a light beam in a plasma of
density n, then T|=(l-n/ncrit)1/2, where ncrit is the critical density. When the probe beam
collinearly samples the RPW, the photons that propagate in higher density plasma will
tend to propagate slower while the ones that propagate in lower density plasma will
tend to propagate faster, according to the group velocity of an electromagnetic wave in
a plasma: vg=cr|. Equivalently, from the wave-wave interaction point of view, the
phase velocity of the probe beam, vp=c/T|, is changing according to the variation of T|.
Thus, the result of a light wave probing a forward moving relativistic plasma wave is a
phase and amplitude modulation of the probe beam, which is actually the collinear TS
process.7 The way this modulation is exploited to provide information about the plasma
wave is shown in Fig. 1.
The scattered beam is directed onto a grating to transform the phase and amplitude
modulation into sidebands. The intensity of the sidebands is related to the TS
efficiency and is approximately 10"8-10"10 of the intensity of the probe pulse, as we
mentioned before. The separation between the sidebands is given by the period of the
modulation which is -1 ps in our experiment. That makes the separation between the
red and blue shifted fundamentals (1-st order sidebands) approximately 16 A. The
detection and measurement of these very weak sidebands, situated only 8 A away from
the intense probe light, represent the difficult tasks of this experiment.
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Amplitude modulation is
FIGURE 1. Production of sidebands from the phase and amplitude modulated probe beam after it
sampled the plasma wave. For resonant excitation of RPW at n=1016 cm"3, the sideband separation is
-16 A.
EXPERIMENT
In our experiment, the excitation of the RPW is produced by beating the 10P20 CO2
line (10.6 jim) with the 10R16 CO2 line (10.3 jim). The pulse has a FWHM of
approximately 100 ps and contains 100 J of energy, thus bringing the CO2 pulse at the
TW level of power.8 The energy is generally split between the two CO2 lines in a ratio
10.6 jim to 10.3 jim of approximately 3:1. The beam is focused into an interaction
chamber with an off-axis paraboloid (OAP) with a focal length f=152 mm. The
focussing given by the OAP can be varied from f/1.5 to f/4 by limiting the transverse
extent of the beam with an iris. The plasmas are produced by tunneling ionization of
the filling gas. The experiments were done mostly with f/3 focussing with ionization
intensities up to 1015 W/cm2. This intensity is high enough to produce plasmas with a
FWHM radial extent of 100-120 jim and over 2 mm in length in hydrogen.
The plasma density is given by the initial pressure of the filling gas. The pressure
was monitored with a capacitance manometer with an accuracy of 5% of the reading.
The plasma density is also directly inferred from large-angle TS measurements.9 By
correlating this TS data with the pressure readings we can conclude that all H2
molecules are being ionized by the laser pulse and the plasma density can be changed
by varying the filling gas pressure.
The experimental setup is shown (top view) in Fig. 2. The plasmas are produced at
the interaction point (IP) by the CO2 laser pulse. The coupling of the probe beam onto
the plasma waves is done through a hole in the OAP. The probe pulse is produced by a
frequency-doubled, commercial NdiYAG laser (DCR-11, Quanta-Ray). This laser
provides a linearly-polarized Q-switched pulse having approximately 50 ml of energy
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in -2ns. The wavelength was measured to be ?tprobe=532.1 nm (green) using a Ne lamp
(Oriel Instruments). Other spectral measurements showed that the bandwidth of the
laser pulse is small enough to allow the detection of any sideband situated more than 23 A away from ?lprobe. The pointing stability was measured to be better than 0.2 mrad
when the operating temperature is constant, a fact that, combined with the good
pointing stability of the CO2 pulse, ensured that the scattering process takes place every
shot. Furthermore, to avoid any chance of accidental misalignment during operation, an
on-line beam position monitor was installed by collecting the "leak" through a highreflective mirror.
_ Vacuum Chamber
Etalon (high-power filter)
Beam Dump
FIGURE 2. Experimental setup.
The probe pulse is focussed at the IP with a f/30 lens. The sidebands are produced
in the plasma and, after the IP, the scattered light (the signal) and the probe beam (the
stray light) are recollimated with another lens and transported out of the interaction
chamber. The signal has to be separated from the stray light in order to frequency and
time-resolve the sidebands. Due to the very low efficiency of the TS process,
10""8 to 10~10, a special spectral separator consisting of an etalon, a spatial filter and a
grating has been built. These optical elements and the way they are positioned in the
experimental setup are shown in Fig. 2.
The 1.4-A half-width etalon (air-spaced) is designed to transmit the now unwanted
stray light and to reflect as much as possible of the sidebands. Having a thickness of
50 um, the etalon reflects all frequencies in a range of ±27 A. This means that the
etalon can reflect approximately 95% of the first three harmonics on each side of the
spectrum, both blue and red shifted sidebands. In this experiment though, only the
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wavelengths in ~44-A range can be transported to the detection system with no
significant losses. Due to stray interference effects related to the external green antireflection coated surfaces, the etalon also reflects -10% of the stray light. For this
reason, a 3-pass geometry has been employed to reduce the level of 532-nm light 3
orders of magnitude, not shown in Fig. 2 for simplicity.
A holographic grating is used after the etalon in one-pass only in order to further
decrease the stray light level significantly, see Fig. 2. The linear dispersion of the
grating is 3.5 A/mm in the focal plane of the lens (f=l m) that is used to focus the 532nm stray light and dump it on a small set of razor blades. The thickness of the dump
can be adjusted to pass the sidebands and other frequencies of interest. Overall, this
"filtering grating" introduces an additional attenuation factor for the stray light of ~10~6.
The combined action of the etalon and grating reduces the stray light a factor of 10~9. It
has to be mentioned that a spatial filter is used in between these two optical elements,
see Fig. 2. The role of this filter is to transmit the signal and to block the stray light that
is generated on all the optical surfaces up to that point. This light propagates on a
different path than the 532-nm light and, if not stopped, would reduce the capability of
the grating to filter out the stray light 103 times.
After the filtering process mentioned above, the power of the probe beam becomes
-2 mW, including losses. At this level of power, no damage can be done to the streak
camera, which is used as a detector in combination with an imaging spectrometer. The
spectral resolution of this detection system is Av=5.3 1010 Hz and the temporal
resolution is -100 ps. The temporal resolution is given by the uncompensated gratings,
one used as a filter, and the other one used in the spectrometer. The dispersion planes
of these two gratings are orthogonal and so the "stretching" effect induced by each of
them is combined to give the 100 ps resolution. The stretching effect was investigated
with a 20 ps visible pulse that became -100 ps while propagating on the same path as
the scattered light. The temporal resolution depends only on the total number of
grooves covered by the beam on each grating, and thus this resolution can be improved
2-3 times just by limiting the size of the beam on these gratings, provided there is
enough signal to be detected.
EXPERIMENTAL RESULTS
RPW are generated and detected in plasmas produced in He, Hydrogen and Ar at
densities varying from 1/5 n^ up to 14 nres. A typical time-resolved spectrum of the
scattered light is presented in Fig. 3.
The resonant pressure in helium (He1+) is approximately 300 mTorr. Figures 3(a)
and 3(b) show the time-resolved spectrum of the scattered light at resonance and 6
times above the resonance, respectively. The red and the blue shifted sidebands have
similar intensities, as expected from the collinear TS modulation process presented in
Fig. 1. In both Fig. 3(a) and Fig. 3(b) the sidebands are each separated by
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approximately 8.1 A, which is exactly the wavelength shift that is associated with Aco,
the beat pattern of the CO2 laser pulse. This means that, even if the waves are excited
very much off-resonance, the only response that the plasma is producing is the one that
carries the signature of the excitation source, Aco. In Fig. 3(a) only the fundamentals of
the RPW are observable while in Fig. 3(b), both the fundamentals and second
harmonics are visible (on the red side even the third harmonic). This is due to the
different conditions of excitation, more powerful pump pulses being used in the nonresonance excitation (approximately 2 times more). Another part of the explanation is
that, at higher plasma densities there are more scatterers that participate in the TS
process. This can be seen better in Eq. (3) where Sn is the only plasma density related
quantity. If 8 is the wave amplitude, then Sn=e n. For equal amplitude waves there is
more scattering at higher densities.
531.0531.5532.0532.5 -
533.0-
500
50
1000
Time (ps)
500
200 350 500 650
Intensity (counts)
50
(a)
1500 2500
Time (ps)
70
90
Intensity (counts)
(b)
FIGURE 3. Red and blue shifted sidebands generated by collinear Thomson scattering of the 532.1-nm
probe pulse on the relativistic plasma wave excited in He at (a) 322 mTorr and (b) 1.7 Torr; the intensity
of the CO2 pulse was approximately 1015 W/cm2 in (a) and 1.8 1015 W/cm2 in (b).
Temporally, the sidebands are short lived, their duration being approximately
100 ps (FWHM). As we mentioned before, the temporal resolution of the TS setup
does not allow measurements faster than 100 ps. Therefore, it can only be concluded
from Fig. 3 that the duration of the plasma wave is shorter or comparable to the
duration of the CO2 pulse. The plasma wave decays significantly, below the detection
threshold for the streak camera, after the action of the infrared pulse on the plasma is
over. Also, the sidebands are all excited at the same moment in time, the slight
temporal tilt that can be observed in both Fig. 3(a) and Fig. 3(b) being just due to the
difference in the optical path induced by the filtering system for different frequencies.
As we mentioned before, the main information that can be extracted from the
collinear TS measurements is the amplitude of the wave. By rearranging Eq. (3) to
extract 8, the wave amplitude, we obtain:
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£ =
(4)
where farea is the aspect ratio between the transverse size of the CO2 and the probe
beam. This formula, applied to the spectra similar to those showed in Figures 3 (a) and
3(b), gives the value of the wave amplitude for each individual shot. This information,
obtained for H2, is presented in Fig. 4.
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10
12
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14
Normalized plasma density
FIGURE 4. Wave amplitude versus the normalized plasma density from TS data for all shots in H2
taken with different intensities of the CO2 pump pulses.
There are three sets of data shown in Fig. 4, depending on the intensity of the CO2
pulse at which the shots were taken. The amplitude of the waves excited at n»nres is
around 10 times smaller than the amplitude of the RPWs excited at resonance. The
intensity of the scattered light at n»nres is though higher than the intensity recorded in
the TS sidebands produced when rrai^, much like the data obtained in He shown in
Fig. 3(a) and Fig. 3(b). As we mentioned before, the difference comes mainly from the
plasma density factor, n, present in Eq. (4).
We can conclude from the experimental data that RPW can be excited even if the
resonance condition is not met and these waves resemble the profile of the beat pattern
of the pump pulse. Their frequency is Aco and their wavenumber is Ak, the difference
between kx and k2, the wavenumbers associated with cOj and co2 of the pump laser. The
amplitude of these non-resonantly excited waves is though much smaller (10-12 times)
than the amplitude of the RPWs excited at resonance, when pump pulses with similar
intensities are used.
Excitation of RPW at densities high above n^ was also reported in a recent solidstate laser-based PBWA experiment10 where a 1-um pump laser pulse was focused to
~10um spot size to achieve intensities above 1018W/cm2. RPWs were measured
through self-scattering of the 1-um pulse. The scattered light was recorded to be most
intense at plasma densities 12 times above resonance. The explanation for this effect is
based on the transverse ponderomotive force of the laser pulse which is powerful
enough to expel the plasma from the focal region and therefore decrease the plasma
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density from n=12 n^ to n=nj.es where the scattering becomes more intense. Thus, even
if this PBWA experiment is performed at n»nres, significant plasma waves are still
driven at resonance.
In our experiment the effect of the ponderomotive blowout is very much reduced
because the CO2 pulse is 3 to 4 orders of magnitude less intense (our normalized laser
intensity a~0.15 versus a~l in Ref. 10) and is focussed to a larger spot size. Moreover,
non-resonant excitation of RPW is recorded in a slightly modified version of our
experiment where the CO2 pulse is focussed with a f/18 lens to a larger spot size of
-350 um (maximum intensity of the pulse is 2.5 1014 W/cm2). It has to be mentioned
that the RPW are also measured using collinear TS but in a slightly different geometry
than the one shown in Fig. 2. The CO2 laser pulse was focussed with a f/18 focussing
lens to a spot size (FWHM) of -350 um and we used the same collinear TS technique
to detect the RPWs. The sidebands recorded in this experiment where the transverse
extent of the RPW is much larger than c/cop are very similar with the ones presented in
Fig. 3. Therefore, it can be concluded that the RPWs excited by the CO2 pulse at
n»nres exist for the duration of the pump pulse in a plasma with density close to n.
Even if the amplitude of the non-resonantly excited RPW is small, their potential to
accelerate electrons is still defined by the associated amplitude and phase of the
longitudinal electric field. In a 1-D model, the amplitude of the electric field defined
by a RPW excited at the resonant plasma density, n^, is:
Eres=0.96erraA/n^
(5)
3
where Eres is in V/cm and n^ in cm" . In the case of non-resonant excitation at a plasma
density n, which is N times above n^, the longitudinal electric field amplitude is:
E = 0.966-R^
(6)
The information resulting from Eqs. (5) and (6) is that, for example, a 1% wave
excited at 10 times above n^ produces the same acceleration field as a 10% wave
driven at nres. So, even if the amplitude of the RPW excited in the non-resonant
conditions is small, their longitudinal electric field can be high enough to accelerate
electrons with gradients of ~GeV/m.
Eqs. (5) and (6) are though valid only in the 1-D model. In reality the relationship
between 8, the wave amplitude, and E, the associated longitudinal electric field is more
complicated. In an attempt to better understand and study the excitation process in both
the resonant and non-resonant conditions, we performed 2-D PIC simulations using the
code turbo WAVE. The code treats electrons as particles that are subject to the
ponderomotive force of the laser pulses impinging on them. It includes ionization
routines for the gases that are investigated, ionization induced refraction and it is
relativistic. The results from these simulations are presented in the next section.
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SIMULATIONS
The simulations are performed on a cluster of 3 G4 Apple computers. Usually, 16
particles are placed in a grid of 1024x256 cells with O.lc/cop size per cell. The code is
set to run for 10000 steps with 0.1/cop time step, which makes the running time to be
-48 hours. The duration (FWHM) of the laser pulses is between 50 ps and 100 ps.
The frequency difference Aco is varied compared to cop in turbo WAVE by adjusting
the laser frequencies and keeping the plasma density constant. All the other parameters,
like the laser pulse rise time, duration, etc., have to also be adjusted accordingly.
Initially, 1-D simulations are performed to verify the scaling law presented in Eq. (6).
The output of the code is given in normalized quantities like Sn/n and E/Ecoldatn, where
E C oidatn is the cold wavebreaking limit at density n, or Ecoldatn(V/cm)=0.96 (n)1/2. To
compare the 1-D simulation results with the theoretical predictions, we first rewrite
Eq. (6) in normalized quantities:
E/Ecoldatn
———
The code performed 10 runs where the density is varied from -0.4 n^ to 16 n^. The
1-D simulation results are presented in Fig. 5. Each dot represents the ratio between
the normalized longitudinal electric field and the wave amplitude versus the square root
of the normalized density for each run. The dotted line is a linear fit for these points
and it resembles Eq. (7) within the error of determining the exact values for the wave
amplitude and the electric field from turbo WAVE. It can be concluded that the code
accurately models the excitation of RPW in the non-resonant regime.
i.
8
•*
3.5
4
,'
•^s
1
I
I'"'
2
--\-~j,*
s?
0
0
0.5
1
1.5
2
2.5
3
Sqrt(n/n )
FIGURE 5. Ratio of normalized longitudinal electric field and wave amplitude versus the squareroot of n/nres from the 1-D simulations with turboWAVE (dots); the dashed line represents a linear fit for
the data points.
Following the good agreement between theory and the 1-D simulations, 2-D runs
are undertaken. In Fig. 6 a snapshot of both the intensity of the laser pulse and the
longitudinal electric field is shown.
For this particular case the density is n=1.3 n^, and a1=a2=0.3, where al and ^ are
the normalized laser intensities for each frequency in the pump pulse. In Fig. 6 (a), the
beat pattern of the CO2 pulse is moving from left to right and the snapshot is taken
approximately when the peak of the pulse passes through the focal plane set at
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x=1.8 mm. In fig. 6(b) the longitudinal electric field associated with the RPW is shown
for the same moment in time. The separation between the beats in Fig. 6(a) is equal to
the wavelength of the RPW represented by its electric field in Fig 6(b), as expected for
a case of resonant excitation of plasma waves.
(b)
FIGURE 6. Results of 2-D H2 simulations with ionization when plasma density n=1.3 n^; snapshot
of the normalized laser intensity (a) and the longitudinal electric field E associated with the RPW (the
laser pulse moves from left to right).
For this particular case the density is n=1.3 n^, and a1=a2=0.3, where al and ^ are
the normalized laser intensities for each frequency in the pump pulse. In Fig. 6 (a), the
beat pattern of the CO2 pulse is moving from left to right and the snapshot is taken
approximately when the peak of the pulse passes through the focal plane set at
x=1.8 mm. In fig. 6(b) the longitudinal electric field associated with the RPW is shown
for the same moment in time. The separation between the beats in Fig. 6(a) is equal to
the wavelength of the RPW represented by its electric field in Fig 6(b), as expected for
a case of resonant excitation of plasma waves.
2-D simulations are also performed at n=7 n^ at the same excitation conditions.
The wavelength of the plasma waves driven off-resonance is still 340 um, in agreement
with the TS experimental results. The major difference between RPWs excited at
n=7 n^ and at resonance is the value of the longitudinal electric field, Ex, which is ~2.5
times lower for the non-resonant case. In the 2-D simulation results, Ex is smaller
(when compared to the 1-D simulations with the same wave amplitude) due to the
appearance of a transverse component, Ey. Therefore, the potential for acceleration of
electrons that is offered by large-amplitude RPW of small transverse extent (< c/cop)
diminishes up to 3 times because of the coupling of these waves into intense transverse
fields, Ey. Eq. (7) is not satisfied anymore in 2-D simulations in which the transverse
extent of the plasma wave is smaller or comparable to c/cop. It has to be mentioned that
no other types of relativistic waves are revealed by the 2-D simulations.
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Data similar to the one shown in Fig. 6 can be extracted from turbo WAVE for
other important quantities like the electron and ion distribution. The wave amplitude
can be determined directly from the electron output of the code for each run that is
performed at parameters close to the experimental ones. The results are compared to
the values of the wave amplitude, 8, calculated from the TS measurements. The
simulation results agree with the experimental values of 8 within 25% for the welldocumented shots and within 40% for the shots where the experimental data was
subject to larger uncertainties. Overall, we can conclude that the 2-D simulations
provide good estimations for the amplitude of the RPWs driven at both resonant and
non-resonant conditions.
The 2-D simulations offer the opportunity to study various phenomena that are
related to the capability of the RPW to accelerate electrons, like relativistic saturation
and ponderomotive blowout.
Relativistic detuning of RPWs11 appears in the 2-D simulations mostly in the focal
region where the intensity of the CO2 pulse is high enough to "overdrive" the plasma
wave at resonance. The phase of the wave slips back compared to the phase of the beat
pattern of the CO2 pulse in some cases up to n rad and the RPW damps significantly.
To illustrate this effect using the output of the code we save the "axis" line, that is, all
the x-cells that have the y-coordinate the middle of the transverse (y) extent for each
time step. Then we build a 2-D matrix by stacking these horizontal lines one on top of
each other, in their time order. In this way we record the change of important quantities
on the "axis" line. Such a 2-D matrix constructed for the normalized laser intensity is
shown in Fig. 7(a) for the run related to Fig. 6 (n=1.3 n^). A similar matrix shows the
variation of the longitudinal electric field on axis in time, see Fig. 7(b). The initial
moment t=0 in Fig. 7 is arbitrarily chosen -10 ps after the laser pulse ionized the gas.
The striations (fringes) visible in Fig. 7(a) represent the beats of the two-frequency CO2
pulse.
O.OQ; .0;i:o ; 0.20
:
No rma iiz & d la ser intensity
•w--
(a)
FIGURE 7. The variations in time of (a) the normalized laser intensity on axis, and (b) the longitudinal
electric field on axis at n=1.3 n,.es.
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The beat pattern moves at the speed of light, which is actually the slope of the
fringes in Fig. 7(a). The "laser fringes" are always straight (they do not curve) because
the plasma does not affect the propagation of the CO2 pulse. The beat pattern excites
the plasma wave through the ponderomotive force and the wavefronts of the wave are
shown in Fig. 7(b). Due to the fact that the laser is more intense in the focal region, the
wave grows faster there and starts to saturate. The manifestation of the RPW saturation
is, first, the curving of the wavefronts, and second, the damping of the wave (as the
RPW goes out-of-phase with the CO2 driver). Both of these phenomena can be
observed in Fig. 7(b) at approximately x=1.8 mm.
The relativistic detuning effect is very important when acceleration of electrons
using these RPW is considered. The electrons are injected into the wave structure at
already relativistic velocities. They will "ride" along one of the "fringes" shown in
Fig. 7(b). The negative effect of relativistic saturation (wavefront curvature) can be
estimated by summing the electric field values along the slope of the speed of light.
Preliminary calculations show that the negative contribution brought in by the
relativistic saturation around the focal region is not significant when compared to the
overall positive acceleration gradient that develops before and after the focus.
A similar study is done for the case where the plasma density is 7 times above
resonance, n=7 n^. The pump pulse has the same intensity, a1=a2=0.3 and the CO2
pulse spot size is similar, w0=50 um. A RPW is excited having a maximum amplitude
8=2%, approximately 10 times smaller than 8 at resonance. The variation of the
longitudinal electric field associated with this RPW on axis is presented in Fig. 8. As
we mentioned before the maximum value of the Ex in this case is -8 MV/cm, 2.5 times
smaller than Ex at resonance.
10-
E§
2(
H
30-
0.25
0.75
x(mm)
-10
-5
0
1.25
5
10
Longitudinal electric field (MV/cm)
FIGURE 8. The variation in time of the longitudinal electric field on axis associated with RPW
driven at n=7 nres. The wavefronts do not curve, the plasma wave follows the beat pattern of the CO2
pulse. The irregular structure that precedes the appearance of the plasma wave is given by the ionization
process.
782
Let us now compare the excitation processes revealed by the simulations at
resonance (i^LSn^) and off-resonance (n=7 nres). One difference is that, when
n»nres, the relativistic saturation phenomenon does not exist anymore. The plasma
wave develops as the pump pulse dictates, or, in other words, the pump and the wave
are always in phase. That can be observed in Fig. 8 where the curving of the
wavefronts does not appear anymore, as it happened with the excitation at resonance.
Another important difference between these two regimes of excitation is related to
the ponderomotive blowout of the plasma. The transverse ponderomotive force of the
CO2 laser is pushing the electrons outwards creating a transverse charge imbalance.
The ions are slowly moving away from the axis to keep up with electrons and thus the
plasma density on axis decreases at a rate determined by the intensity of the pump
pulse. When RPW are excited at resonance, it is very important to maintain the plasma
density within few percent of n^.12 This is very difficult to do when strongly focussed,
intense pump pulses are used due to the above mentioned blowout phenomenon. When
RPW are excited at densities much above n^, the plasma wave can still be driven even
if the density is constantly decreasing. This positive effect is observed in our
simulations at n=10 n^ when intense pump pulses are used. Even if the plasma density
decreases more than 2 times from its initial value, RPWs are still excited for the time
the pump pulse is present in the plasma.
Excitation of RPW at n»nres has one more advantage if injection of phase-locked
electron bunches is considered. The wavelength of the non-resonantly excited plasma
wave, A,, is given by the beat pattern of the pump pulse. For example, in this PBWA
experiment, A=340 um, large enough to allow future injection of short (<1 ps) electron
bunches that will be phase-locked with the RPW excited at n=14 nres. If the laser
frequencies are changed to provide resonant excitation at the same density, a -10 times
higher amplitude wave can be excited with the same laser power. At the same time the
beat pattern of the CO2 pulse will drive a plasma wave with A=90 pm, which makes
phase locking of externally injected electron bunches (<250 fs) a much more difficult
task. The overall goal is to excite a high-amplitude, large-wavelength RPW in a highdensity plasma.
One way to increase the amplitude of non-resonantly excited RPW is to use the
relativistic optical guiding process. The power threshold to achieve it is Pc=17(co/cop)2
GW.13 On one hand, relativistic guiding increases the laser power that is driving the
wave and, on the other, is making the interaction length longer. At the parameters for
this experiment, at resonance, n^lO16 cm"3, the critical power would be 15 TW which
is 15 times more than the maximum power achieved, 1 TW. This threshold decreases
to -5 TW at n=10 n^. Simulations done so far at this density with 5 TW laser pulses
show that the laser intensity on axis increases -3 times and a plasma channel of 50%
783
the initial density is formed. In this channel non-resonant RPW with amplitudes of up
to 25% are excited and acceleration fields of -30 MV/cm are obtained. The plasma
wave wavelength is still 340 um.
Excitation of high amplitude RPW at n»nres is also possible in capillary discharge
plasma channels that are easily available at densities higher than 1017 cm"3. On the
experimental side, high-efficiency, stable guiding of TW pulses has already been
shown in these channels.14 Turbo WAVE simulations with preformed plasma channels
represent the next step in the study of generation of RPW in such conditions.
CONCLUSIONS
We have studied the beat-wave excitation process of RPWs at resonance and offresonance. Experimentally, the RPWs driven by a TW CO2 laser pulse are detected
using collinear Thomson scattering by sampling the plasmas with a 532-nm probe
pulse. The red and the blue shifted sidebands (up to the 3-rd harmonic) are recorded in
a wide range of plasma densities (1015cm~3 to 1017cm"3). The experimentally measured
amplitude of the plasma wave, 8, is maximum 10% at resonance and maximum 1% offresonance. These values are compared to the predictions from 2-D simulations and the
results agree quite well with the experiment. Furthermore, it is shown that, even if 8 is
small when n»nres, the longitudinal electric field associated with the RPW is high
enough to accelerate electrons due to the increased number of electrons that are part of
the wave. The simulations reveal the fact that, in the case of non-resonant excitation at
n»nres, negative phenomena like relativistic saturation and ponderomotive expulsion
of plasma do not affect the RPW.
ACKNOWLEDGEMENTS
The authors wish to acknowledge the helpful support of Dr. D. Gordon for the
simulations done with turbo WAVE. This work was supported by U. S. DoE Grant No.
DE-FG03-92-ER40727.
REFERENCES
1. T. Tajima and J. M. Dawson, "Laser electron accelerator," Phys. Rev. Lett. 43, 267-270 (1979).
2. C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, "Relativistic plasma-wave excitation by
collinear optical mixing," Phys. Rev. Lett. 54, 2343-2346 (1985).
3. A. Dyson, A. E. Dangor, A. K. L. Dymoke-Bradshaw, T. Ashfar-Rad, P. Gibbon, A. R. Bell, C. N.
Danson, C. B. Edwards, F. Amiranoff, G. Matthieusent, S. J. Karttunen, and R. R. E. Salomaa,
784
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
"Observations of relativistic plasma waves excited by a 1.064 um and 1.053 um laser beat," Plasma
Physics and Controlled Fusion 38, 505-525 (1996).
F. Amiranoff, M. Laberge, J. R. Marques, F. Moulin, E. Fabre, B. Cros, G. Matthieussent, P.
Benkheiri, F. Jacquet, J. Meyer, P. Mine, C. Stenz, P. Mora, "Observation of modulational instability
in Nd-laser beat-wave experiments," Phys. Rev. Lett. 68, 3710-3713 (1992).
A. K. Lal, D. Gordon, K. Wharton, C. E. Clayton, K. A. Marsh, W. B. Mori, C. Joshi, M. J. Everett,
and T. W. Johnston, "Spatio-temporal dynamics of the resonantly excited relativistic plasma wave
driven by a CO2 laser," Phys. Plasmas 4, 1434-1447 (1997).
R. E. Slusher and C. M. Surko, "Study of density fluctuations in plasmas by small-angle CO2 laser
scattering," Phys. Fluids 23, 472 (1980).
E. Esarey, A. Ting, and P. Sprangle, "Frequency shifts induced in laser pulses by plasma waves,"
Phys. Rev. A 42, 3526-3531 (1990).
S.Ya. Tochitsky, C. Filip, R. Narang, C.E. Clayton, K.A. Marsh, and C. Joshi, "Efficient shortening
of self-chirped picosecond pulses in a high-power CO2 amplifier," Opt. Lett. 26, 813-815 (2001).
R. Narang, C. V. Filip, S. Ya. Tochitsky, D. Gordon, C. E. Clayton, K. A. Marsh, and C. Joshi,
"Characteristics of plasmas produced by double beat wave interaction in the Neptune facility at
UCLA.," Proc. 2001 Particle Accelerator Conf., eds. P. Lucas, S. Webber, 5, 3996-3998 (2001).
B. R. Walton, Z. Najmudin, M-S Wei, C. Marie, k. Krushelnick, A. E. Dangor, J. Clarke, M. J.
Poulter, C. Hernandez-Gomez, S. Hawkes, D. Neely, C. N. Danson, J. L. Coolier, S. Fritzler, and V.
Malka, " Short pulse laser beatwave experiments using the VULCAN laser facility," Bull. Am. Phys.
Soc. 46, 90 (2001).
M. N. Rosenbluth and C. S. Liu, "Excitation of plasma waves by two laser beams," Phys. Rev. Lett.
29, 701-705 (1972).
C. M. Tang, P. Sprangle, and R. N. Sudan, "Dynamics of the space-charge waves in the laser beat
wave accelerator," Phys. Fluids 28, 1974-1983 (1985).
P. Sprangle, C. M. Tang, and E. Esarey, "Relativistic self-focusing of short-pulse radiation beams in
plasmas," IEEE Trans. Plasma Sci. PS-15, 145-153 (1987).
A. Zigler, Y. Ehrlich, C. Cohen, T. Krall, and P. Sprangle, "Optical guiding of high-intensity laser
pulses in a long plasma channel formed by a slow capillary discharge.," Journal of the Optical
Society of America B, 13, 68-71 (1996).
785